On the Gram-Schmidt orthonormalizatons of subsystems of Schauder systems

Robert E. Zink

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Subsystems of the Schauder system whose orthonormalizations are Schauder bases for $L^{p} [0, 1]$

Robert E. Zink (1989)

Colloquium Mathematicae

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Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces

Aydin Sh. Shukurov (2012)

Colloquium Mathematicae

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A necessary condition for Kostyuchenko type systems and system of powers to be a basis in $L_{p}$ (1 ≤ p < +∞) spaces is obtained. In particular, we find a necessary condition for a Kostyuchenko system to be a basis in $L_{p}$ (1 ≤ p < +∞).

Addendum to "Necessary condition for Kostyuchenko type systems to be a basis in Lebesgue spaces" (Colloq. Math. 127 (2012), 105-109)

Aydin Sh. Shukurov (2014)

Colloquium Mathematicae

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It is well known that if φ(t) ≡ t, then the system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is not a Schauder basis in L₂[0,1]. It is natural to ask whether there is a function φ for which the power system ${φ ⁿ (t)}_{n = 0}^{\infty}$ is a basis in some Lebesgue space $L_{p}$ . The aim of this short note is to show that the answer to this question is negative.

Three-space problems and bounded approximation properties

Wolfgang Lusky (2003)

Studia Mathematica

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Let ${R ₙ}_{n = 1}^{\infty}$ be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an $ℒ_{p}$ -space, then both X and A have bases. We apply these results to show that the spaces $C_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset C ()$ and $L_{Λ} = \bar{s p a n} z^{k} : k \in Λ \subset L ₁ ()$ have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.

Uniqueness of unconditional basis of $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ , 0 < p < 1

F. Albiac, C. Leránoz (2002)

Studia Mathematica

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We prove that the quasi-Banach spaces $ℓ_{p} (c ₀)$ and $ℓ_{p} (ℓ ₂)$ (0 < p < 1) have a unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss and Tzafriri have previously proved that the same is true for the respective Banach envelopes $ℓ ₁ (c ₀)$ and ℓ₁(ℓ₂). They used duality techniques which are not available in the non-locally convex case.

A basis of Zₘ

Min Tang, Yong-Gao Chen (2006)

Colloquium Mathematicae

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Let $σ_{A} (n) = | (a, a^{'}) \in A ² : a + a^{'} = n |$ , where n ∈ N and A is a subset of N. Erdős and Turán conjectured that for any basis A of order 2 of N, $σ_{A} (n)$ is unbounded. In 1990, Imre Z. Ruzsa constructed a basis A of order 2 of N for which $σ_{A} (n)$ is bounded in the square mean. In this paper, we show that there exists a positive integer m₀ such that, for any integer m ≥ m₀, we have a set A ⊂ Zₘ such that A + A = Zₘ and $σ_{A} (n ̅) \leq 768$ for all n̅ ∈ Zₘ.

On the non-equivalence of rearranged Walsh and trigonometric systems in $L_{p}$

Aicke Hinrichs, Jörg Wenzel (2003)

Studia Mathematica

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We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in $L_{p}$ for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

Unconditionality of general Franklin systems in $L^{p} [0, 1]$ , 1 < p < ∞

Gegham G. Gevorkyan, Anna Kamont (2004)

Studia Mathematica

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By a general Franklin system corresponding to a dense sequence = (tₙ, n ≥ 0) of points in [0,1] we mean a sequence of orthonormal piecewise linear functions with knots , that is, the nth function of the system has knots t₀, ..., tₙ. The main result of this paper is that each general Franklin system is an unconditional basis in $L^{p} [0, 1]$ , 1 < p < ∞.

Matrix subspaces of L₁

Gideon Schechtman (2013)

Studia Mathematica

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If $E = e_{i}$ and $F = f_{i}$ are two 1-unconditional basic sequences in L₁ with E r-concave and F p-convex, for some 1 ≤ r < p ≤ 2, then the space of matrices $a_{i, j}$ with norm $| | a_{i, j} {| |}_{E (F)} = | | \sum_{k} | | \sum_{l} a_{k, l} f_{l} | | e_{k} | |$ embeds into L₁. This generalizes a recent result of Prochno and Schütt.

Continuity of halo functions associated to homothecy invariant density bases

Oleksandra Beznosova, Paul Hagelstein (2014)

Colloquium Mathematicae

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Let be a collection of bounded open sets in ℝⁿ such that, for any x ∈ ℝⁿ, there exists a set U ∈ of arbitrarily small diameter containing x. The collection is said to be a density basis provided that, given a measurable set A ⊂ ℝⁿ, for a.e. x ∈ ℝⁿ we have $l i m_{k \to \infty} 1 / | R_{k} | \int_{R_{k}} χ_{A} = χ_{A} (x)$ for any sequence $R_{k}$ of sets in containing x whose diameters tend to 0. The geometric maximal operator $M_{}$ associated to is defined on L¹(ℝⁿ) by $M_{} f (x) = s u p_{x \in R \in} 1 / | R | \int_{R} | f |$ . The halo function ϕ of is defined on (1,∞) by $ϕ (u) = s u p 1 / | A | | x \in ℝ ⁿ : M_{} χ_{A} (x) > 1 / u | : 0 < | A | < \infty$ and on [0,1] by ϕ(u) = u. It is shown...

A basis of ℤₘ, II

Min Tang, Yong-Gao Chen (2007)

Colloquium Mathematicae

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Given a set A ⊂ ℕ let $σ_{A} (n)$ denote the number of ordered pairs (a,a’) ∈ A × A such that a + a’ = n. Erdős and Turán conjectured that for any asymptotic basis A of ℕ, $σ_{A} (n)$ is unbounded. We show that the analogue of the Erdős-Turán conjecture does not hold in the abelian group (ℤₘ,+), namely, for any natural number m, there exists a set A ⊆ ℤₘ such that A + A = ℤₘ and $σ_{A} (n ̅) \leq 5120$ for all n̅ ∈ ℤₘ.

Decomposition systems for function spaces

G. Kyriazis (2003)

Studia Mathematica

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Let $Θ : = θ_{I}^{e} : e \in E, I \in D$ be a decomposition system for $L ₂ (ℝ^{d})$ indexed over D, the set of dyadic cubes in $ℝ^{d}$ , and a finite set E, and let $Θ ̃ : = Θ ̃_{I}^{e} : e \in E, I \in D$ be the corresponding dual functionals. That is, for every $f \in L ₂ (ℝ^{d})$ , $f = \sum_{e \in E} \sum_{I \in D} ⟨ f, Θ ̃_{I}^{e} ⟩ θ_{I}^{e}$ . We study sufficient conditions on Θ,Θ̃ so that they constitute a decomposition system for Triebel-Lizorkin and Besov spaces. Moreover, these conditions allow us to characterize the membership of a distribution f in these spaces by the size of the coefficients $⟨ f, Θ ̃_{I}^{e} ⟩$ , e ∈ E, I ∈ D. Typical examples of such decomposition...

General Haar systems and greedy approximation

Anna Kamont (2001)

Studia Mathematica

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We show that each general Haar system is permutatively equivalent in $L^{p} ([0, 1])$ , 1 < p < ∞, to a subsequence of the classical (i.e. dyadic) Haar system. As a consequence, each general Haar system is a greedy basis in $L^{p} ([0, 1])$ , 1 < p < ∞. In addition, we give an example of a general Haar system whose tensor products are greedy bases in each $L^{p} ({[0, 1]}^{d})$ , 1 < p < ∞, d ∈ ℕ. This is in contrast to [11], where it has been shown that the tensor products of the dyadic Haar system are not greedy bases...

Crystal bases for the quantum queer superalgebra

Dimitar Grantcharov, Ji Hye Jung, Seok-Jin Kang, Masaki Kashiwara, Myungho Kim (2015)

Journal of the European Mathematical Society

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In this paper, we develop the crystal basis theory for the quantum queer superalgebra $U_{q} (𝔮 (n))$ . We define the notion of crystal bases and prove the tensor product rule for $U_{q} (𝔮 (n))$ -modules in the category $𝒪_{int}^{\geq 0}$ . Our main theorem shows that every $U_{q} (𝔮 (n))$ -module in the category $𝒪_{int}^{\geq 0}$ has a unique crystal basis.

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

Zdeněk Dušek (2015)

Czechoslovak Mathematical Journal

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Let $(M, g)$ be a 4-dimensional Einstein Riemannian manifold. At each point $p$ of $M$ , the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor $R$ at $p$ . In this basis, up to standard symmetries and antisymmetries, just $5$ components of the curvature tensor $R$ are nonzero. For the space of constant curvature, the group $O (4)$ acts as a transformation group between ST bases at $T_{p} M$ and for the so-called 2-stein curvature tensors, the group $Sp (1) \subset SO (4)$ acts as a transformation...

Normal number constructions for Cantor series with slowly growing bases

Dylan Airey, Bill Mance, Joseph Vandehey (2016)

Czechoslovak Mathematical Journal

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Let $Q = {(q_{n})}_{n = 1}^{\infty}$ be a sequence of bases with $q_{i} \geq 2$ . In the case when the $q_{i}$ are slowly growing and satisfy some additional weak conditions, we provide a construction of a number whose $Q$ -Cantor series expansion is both $Q$ -normal and $Q$ -distribution normal. Moreover, this construction will result in a computable number provided we have some additional conditions on the computability of $Q$ , and from this construction we can provide computable constructions of numbers with atypical normality properties. ...

Every separable Banach space has a basis with uniformly controlled permutations

Paolo Terenzi

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There exists a universal control sequence ${p ̅ (m)}_{m = 1}^{\infty}$ of increasing positive integers such that: Every infinite-dimensional separable Banach space X has a biorthogonal system xₙ,xₙ* with ||xₙ|| = 1 and ||xₙ*|| < K for each n such that, for each x ∈ X, $x = \sum_{n = 1}^{\infty} x_{π (n)} * (x) x_{π (n)}$ where π(n) is a permutation of n which depends on x but is uniformly controlled by ${p ̅ (m)}_{m = 1}^{\infty}$ , that is, $n_{n = 1}^{m} \subseteq π {(n)}_{n = 1}^{p ̅ (m)} \subseteq n_{n = 1}^{p ̅ (m + 1)}$ for each m.

The postage stamp problem and arithmetic in base $r$

Amitabha Tripathi (2008)

Czechoslovak Mathematical Journal

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Let $h, k$ be fixed positive integers, and let $A$ be any set of positive integers. Let $h A : = {a_{1} + a_{2} + \dots + a_{r} : a_{i} \in A, r \leq h}$ denote the set of all integers representable as a sum of no more than $h$ elements of $A$ , and let $n (h, A)$ denote the largest integer $n$ such that ${1, 2, ..., n} \subseteq h A$ . Let $n (h, k) : = {max}_{A} : n (h, A)$ , where the maximum is taken over all sets $A$ with $k$ elements. We determine $n (h, A)$ when the elements of $A$ are in geometric progression. In particular, this results in the evaluation of $n (h, 2)$ and yields surprisingly sharp lower bounds for $n (h, k)$ , particularly for $k = 3$ .

The basis property in $L_{p}$ of the boundary value problem rationally dependent on the eigenparameter

N. B. Kerimov, Y. N. Aliyev (2006)

Studia Mathematica

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We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in $L_{p}$ of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L₂ we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness...

Second derivatives of norms and contractive complementation in vector-valued spaces

Bas Lemmens, Beata Randrianantoanina, Onno van Gaans (2007)

Studia Mathematica

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We consider 1-complemented subspaces (ranges of contractive projections) of vector-valued spaces $ℓ_{p} (X)$ , where X is a Banach space with a 1-unconditional basis and p ∈ (1,2) ∪ (2,∞). If the norm of X is twice continuously differentiable and satisfies certain conditions connecting the norm and the notion of disjointness with respect to the basis, then we prove that every 1-complemented subspace of $ℓ_{p} (X)$ admits a basis of mutually disjoint elements. Moreover, we show that every contractive projection...

On bases, compactness and weak convergence in the Banach space $A_{p}$

J. T. Marti (1971)

Colloquium Mathematicae

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Explicit extension maps in intersections of non-quasi-analytic classes

Jean Schmets, Manuel Valdivia (2005)

Annales Polonici Mathematici

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We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces $_{()} ({[- 1, 1]}^{r})$ ; (b) there is no continuous linear extension map from $Λ_{()}^{(r)}$ into $_{()} (ℝ^{r})$ ; (c) under some additional assumption on , there is an explicit extension map from $_{()} ({[- 1, 1]}^{r})$ into $_{()} ({[- 2, 2]}^{r})$ by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].

Optimality of the Width- $w$ Non-adjacent Form: General Characterisation and the Case of Imaginary Quadratic Bases

Clemens Heuberger, Daniel Krenn (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider digit expansions $\sum_{j = 0}^{ℓ - 1} Φ^{j} (d_{j})$ with an endomorphism $Φ$ of an Abelian group. In such a numeral system, the $w$ -NAF condition (each block of $w$ consecutive digits contains at most one nonzero) is shown to minimise the Hamming weight over all expansions with the same digit set if and only if it fulfills the subadditivity condition (the sum of every two expansions of weight $1$ admits an optimal $w$ -NAF). This result is then applied to imaginary quadratic bases, which are used for scalar...

On sentences provable in impredicative extensions of theories

Zygmunt Ratajczyk

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CONTENTS0. Introduction.......................................................................... 51. Preliminaries............................................................................... 72. Basic facts to be used in the sequel....................................... 113. Predicates OD(.,.) and CL(.,.).................................................... 174. Predicate Sels............................................................................. 185. Strong $\sum_{n}^{1}$ -collection...........................................................